Part 1
Consider a variant of electoral competition on the line that captures features of the US presidential election. Voters are divided between two states. State 1 has more electoral college votes than does state 2. The winner is the candidate who obtains the most electoral college votes.
Denote by mi the median's favorite position among the citizens of state i, for i = 1, 2; assume that m2 < m1. Each of two candidates chooses a single position. Each citizen votes non-strategically for the candidate whose position is closest to her favorite position and randomize 50-50 when indifferent between candidates. The candidate who wins a majority of the votes in a state obtains all the state's electoral votes; if for some state the candidates obtain the same number of votes, they each obtain half of the electoral college votes of that state.
Find all NE of the strategic game modeled here. What does the set of NE imply about the role of small and large states in elections? In your answer assume that there is an even number of electoral votes in each district and that if candidates tie in the electoral college then each wins the election with probability 1/2.
Part 2
Consider another variant of electoral competition on the line. This time suppose the candidates, like citizens, only care about policy outcomes and not about winning per se. There are two candidates, each who has a favorite position. Suppose that the candidates' utility for an alternative is decreasing in its distance from their ideal policy. Also assume that the favorite policy of candidate 1 (the Democrat) is less than the median
(m) and the favorite policy of candidate 2 (the Republican) is greater than the median. Also assume if the candidates tie then the policy splits the difference between their policy positions, 1/2(x1+x2). Find all of the Nash equilibria for this game.
Part 1:
To find all Nash equilibria in this strategic game, we need to analyze the players' best responses to each other's strategies.
First, let's consider the candidate's decision-making process. Each candidate wants to choose a position that will maximize their chances of winning a state and obtaining its electoral votes.
For a candidate to win a state, they need to have a majority of the votes in that state. This means that each candidate should try to position themselves close to the median's favorite position in each state.
Since the voters vote non-strategically and choose the candidate closest to their preferred position, the candidate who chooses a position closer to the median's favorite position in each state has a higher chance of winning that state.
Now, let's analyze the equilibrium outcomes:
1. If both candidates choose positions that are equally distant from the median's favorite position in both states, then each candidate will receive an equal number of votes in each state. As a result, they will split the electoral votes equally. This outcome is a Nash equilibrium since neither candidate can increase their electoral vote share by unilaterally changing their position.
2. If one candidate chooses a position that is closer to the median's favorite position in one state and the other candidate chooses a position that is closer to the median's favorite position in the other state, then each candidate will win the state they are closer to. This outcome is also a Nash equilibrium since neither candidate can improve their electoral vote share by unilaterally changing positions.
The set of Nash equilibria in this game implies that small states have a greater impact on election outcomes compared to large states. In small states, a candidate can win all the electoral votes by positioning themselves closer to the median's favorite position. In large states, the electoral votes are split between the candidates unless one candidate positions themselves significantly closer to the median's favorite position.
Part 2:
In this variant of electoral competition, the candidates' utility depends on policy outcomes rather than winning per se. Each candidate has a favorite policy position, and their utility decreases as the policy moves away from their preferred position.
Let's analyze the Nash equilibria in this game:
1. If both candidates choose their favorite policy positions, then neither candidate has an incentive to deviate unilaterally. This is a Nash equilibrium since both candidates are playing their best response strategy.
2. If both candidates tie, meaning the policy position chosen is the average of their preferred positions, then neither candidate has an incentive to deviate since they both achieve utility lower than their favorite policy. This is also a Nash equilibrium.
In summary, the Nash equilibria in this game occur when either both candidates play their favorite policy positions or when they tie and choose the average of their preferred positions.